View source for Normal monomorphism

A morphism having the characteristic property of an imbedding of a group (ring) into a group (ring) as a normal subgroup (ideal). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067550/n0675501.png" /> be a [[Category|category]] with zero morphisms. A morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067550/n0675502.png" /> is called a normal monomorphism if every morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067550/n0675503.png" /> for which it always follows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067550/n0675504.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067550/n0675505.png" />, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067550/n0675506.png" />, can be uniquely represented in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067550/n0675507.png" />. The kernel of any morphism (cf. [[Kernel of a morphism in a category|Kernel of a morphism in a category]]) is a normal monomorphism. The converse is not true, in general; however, if cokernels (cf. [[Cokernel|Cokernel]]) of morphisms exist in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067550/n0675508.png" />, then every normal monomorphism turns out to be the kernel of its cokernel. In an [[Abelian category|Abelian category]] every monomorphism is normal. The concept of a normal monomorphism is dual to that of a [[Normal epimorphism|normal epimorphism]].



====Comments====
The above definition is not entirely standard: many authors would define a normal monomorphism to be a morphism which occurs as a kernel. (The term normal subobject is also in use, for an isomorphism class of normal monomorphisms.) In categories without zero morphisms, a substitute for normal monomorphisms is provided by regular monomorphisms, which are those morphisms which occur as equalizers (cf. [[Kernel of a morphism in a category|Kernel of a morphism in a category]]). Every normal monomorphism is regular, but not conversely: in the category of all groups every injective homomorphism is a regular monomorphism, but a normal monomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067550/n0675509.png" /> is an isomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067550/n06755010.png" /> onto a normal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067550/n06755011.png" />. However, in an additive category the concepts of normal monomorphism and regular monomorphism coincide.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Mitchell,   "Theory of categories" , Acad. Press  (1965)  pp. Sect. I.14</TD></TR></table>